The need to visualize and discern structural features in noisy image data (2D or 3D) is ubiquitous to virtually every type of data capture modality and application and is appreciated in many if not most scientific and technical disciplines. In modern biology, in particular, structural visualization is important to the study of chromosome structure and other sub-nuclear component structures of cells, and is made possible in part by the theoretical resolution of state-of-the-art light/electron microscopy (˜50 nm/1.5 Å). At these high resolutions, however, practical limitations posed by the details of sample preparation and the methods used to acquire and process light and electron microscopy (EM) data produce noisy suboptimal images and complicate their interpretation, i.e. visualizing and discerning structural details and locations of features of interest in an image.
For example, the effective resolution of electron microscopy can be limited by the poor specificity of EM stains (non-specific staining). Uranyl acetate is a popular EM stain that forms complexes primarily with phosphates that are localized on DNA, RNA, and phosphoproteins. The consequence is that the acquired image is difficult to interpret, because it is a composite image of superimposed sub-images that contain every stained structure. This is analogous to determining the structure of an object buried in a cloud of nearly the same density material as the object. Interpretation difficulties are compounded if non-specifically stained structures are densely packed, which is typical for condensed chromosomes in a cell nucleus. Furthermore, nonspecific biological noise, detector noise (shot noise), EM microscope alignment uncertainties, and 3D tomographic data reconstruction noise may also contribute to the problem. Due to these factors, 3D EM cell structure cannot be interpreted unambiguously by direct visualization. Light microscopy also has similar and related problems to electron microscopy. Although the stains (especially fluorescent proteins) are spatially very specific compared to EM stains, the reduced specificity of low affinity antibodies may still produce a sizable background. And as in the EM data, densely packed structures compound the interpretational difficulties. Additional sources of noise, such as auto-fluorescence, high background due to pools of unassembled fluorescent proteins, and instrumentation can contribute to produce ambiguous results for 3D light microscopy cell structure studies as well.
Fourier methods (including matched filters) and other data analysis methods, such as correlation and model fitting, have been utilized to bring into view structural details and locations of the features of interest in an image. For Fourier methods in particular, difficulties can occur when there are few repeats of a given signal and much disorder at many angles, which can produce inconclusive results. While Fourier methods provide information about global frequency content (i.e. producing a frequency distribution that is peaked at the most commonly occurring frequencies), it does not provide information about the location of particular frequencies. Thus they are not particularly well adapted to picking out the non-repetitive, multiply oriented structures that comprise most biological data.
In contrast to Fourier methods, wavelet methods are known to be useful for extracting local frequency content, i.e. determining where particular sizes occur independent of how often they occur. A wavelet transform is a convolution of a user-selected wavelet kernel (a shape function) and the data, and can be generally expressed in the 1D case as:
                                          W            ⁡                          (                              x                ,                a                            )                                =                      ∫                                          Ψ                ⁡                                  (                                                            x                      -                                              x                        ′                                                              a                                    )                                            ⁢                              f                ⁡                                  (                                      x                    ′                                    )                                            ⁢                              ⅆ                                  x                  ′                                                                    ,                            Eq        .                                  ⁢                  (          1          )                    where f is the data signal that is convolved with the wavelet kernel function, Ψ. The only physical requirement for Ψ is that a uniform signal of infinite extent produces no correlations, that is,
      ∫                  Ψ        ⁡                  (                                    (                              x                -                                  x                  ′                                            )                        /            a                    )                    ⁢              ⅆ                  x          ′                      =  0.Beyond this restriction, the choice of the wavelet kernel, Ψ, is arbitrary and is tailored usually to the particular problem of interest. The only major exception is for image compression and reconstruction applications (a mature application of wavelets), for which an orthonormal set of wavelets is desirable. A wavelet transform differs from a Fourier transform in the choice of Ψ. The kernel function for a Fourier transform (Ψ(x′)=exp(ix′/a)) has nonzero values that extend over the entire x′ domain. Therefore W for the Fourier transform is only a function of (a). In contrast, the wavelet kernel, Ψ, for the wavelet transform is centered at x and is nonzero over a finite width, (a). It is this property of the wavelet kernel that produces frequency content as a function of spatial location. The wavelet transform, W, can be characterized as a correlation function, i.e. at each position, x, the correlation varies with the wavelet width (a), and is maximal for some value of (a). In other words, the wavelet transform shows how strongly the data correlates or is similar to the wavelet kernel at each location in the data.
Although wavelet theory rests on a well-established theoretical foundation, the mathematical complexity and rigor required for wavelet methods, particularly when analyzing 3D data, often requires specialized mathematical expertise that most scientists and researchers do not possess. An extension of Eq. (1) to 3D can be written as:
                                          W            *                    ⁡                      (                          x              ,              y              ,              z              ,              a              ,              b              ,              c                        )                          =                  ∫                      ∫                          ∫                                                Ψ                  ⁡                                      (                                                                  z                        -                                                  z                          ′                                                                    c                                        )                                                  ⁢                                  Ψ                  ⁡                                      (                                                                  y                        -                                                  y                          ′                                                                    b                                        )                                                  ⁢                                  Ψ                  ⁡                                      (                                                                  x                        -                                                  x                          ′                                                                    a                                        )                                                  ⁢                                  f                  ⁡                                      (                                                                  x                        ′                                            ,                                              y                        ′                                            ,                                              z                        ′                                                              )                                                  ⁢                                  ⅆ                                      x                    ′                                                  ⁢                                  ⅆ                                      y                    ′                                                  ⁢                                                      ⅆ                                          z                      ′                                                        .                                                                                        Eq        .                                  ⁢                  (          2          )                    The complexities in wavelet processing, however, arise from the methods used to evaluate Eq. (2). Since most problems to which wavelets are applied require computational speed, data compression, or data reconstruction, Eq. (2) is evaluated typically using the discrete wavelet transform (DWT), which is based on mathematically complex digital filtering methods. As stated above, image compression and reconstruction applications are best addressed using an orthonormal set of wavelets. Where data compression/reconstruction is not involved, however, the much simpler “continuous wavelet transform” (CWT) is preferably used which is a direct integration of Eq. (2).
In summary, there is a need for a greatly simplified and computationally fast wavelet-based filter and process for filtering image data to find features of a given size in any 2D or 3D dataset, and requiring no special expertise to use. This would preferably entail having a single input parameter which is a characteristic spatial size of the structure of interest, and whose output is a spatial image of correlation strength. In this manner, structural visualization of features and their locations in noisy image data may be achieved, such as condensed chromosomes in a cell nucleus, or images obtained from other applications, such as for example, medical imaging (e.g., bone structure), non-destructive evaluation (e.g., internal cracks and defects), airline baggage scanners, etc.